Question

Evaluate ${\int }_0^{2/3}\frac{dx}{(4+9x^2)}$

Answer 1

$=\frac{1}{9}{\int }_0^{2/3}\frac{dx}{(\frac{4}{9}+x^2)}$

$=\frac{1}{9}{\int }_0^{2/3}\frac{dx}{{\left(\frac{2}{3}\right)}^2+x^2}$

$={\begin{array}{c}\frac{1}{9}\frac{1}{2/3}{\tan }^{-1}\left(\frac{x}{2/3}\right)\end{array}]}_0^{2/3}$
$=\frac{1}{6}[{\tan }^{-1}\left(1\right)-{\tan }^{-1}\left(0\right)]$

$=\frac{1}{6}\frac{\pi }{4}=\frac{\pi }{24}$